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In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces. The snub dodecahedron has 92 faces (the most of the 13 Archimedean solids): 12 are pentagons and the other 80 are equilateral triangles. It also has 150 edges, and 60 vertices. It has two distinct forms, which are mirror images (or "enantiomorphs") of each other. The union of both forms is a compound of two snub dodecahedra, and the convex hull of both forms is a truncated icosidodecahedron. Kepler first named it in Latin as dodecahedron simum in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived equally from either the dodecahedron or the icosahedron, called it snub icosidodecahedron, with a vertical extended Schläfli symbol . ==Cartesian coordinates== Cartesian coordinates for the vertices of a snub dodecahedron are all the even permutations of :(±2α, ±2, ±2β), :(±(α+β/ϕ+ϕ), ±(−αϕ+β+1/ϕ), ±(α/ϕ+βϕ−1)), :(±(α+β/ϕ−ϕ), ±(αϕ−β+1/ϕ), ±(α/ϕ+βϕ+1)), :(±(−α/ϕ+βϕ+1), ±(−α+β/ϕ−ϕ), ±(αϕ+β−1/ϕ)) and :(±(−α/ϕ+βϕ−1), ±(α−β/ϕ−ϕ), ±(αϕ+β+1/ϕ)), with an even number of plus signs, where :α = ξ − 1 / ξ and :β = ξϕ + ϕ2 + ϕ /ξ, where ϕ = (1 + √5)/2 is the golden ratio and ξ is the real solution to ξ3 − 2ξ = ϕ, which is the number: : or approximately 1.7155615. This snub dodecahedron has an edge length of approximately 6.0437380841. Taking the odd permutations of the above coordinates with an even number of plus signs gives another form, the enantiomorph of the other one. Though it may not be immediately obvious, the figure obtained by taking the even permutations with an even number of plus signs is the same as that obtained by taking the odd permutations with an odd number of plus signs. Similarly, the mirror image has either an odd permutation with an even number of plus signs or an even permutation with an odd number of plus signs. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「snub dodecahedron」の詳細全文を読む スポンサード リンク
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